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Is this little toy known ?

Let $E$ be some Banach space, and let $K$ be the closed unit ball of its dual, endowed with the weak-star topology. Also, let $j:E$ $\rightarrow$ $C(K)$ be the natural embedding. Then, if $\pi$ :$E$ $\rightarrow$ $C(K)$ is onto, one must have $\left\Vert \pi-j\right\Vert $ > 1. [Applying this to $E=$ $\ell^{1}$, or to $E=C[0,1]$ (eventually, via Milutin) would be interesting, I think.]

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What's up with this "subjective-argumentative" tag? I question whether it (i) applies here and (ii) would be a useful tag for any post. It seems to me that application of this tag is, itself, rather subjective and argumentative. –  Pete L. Clark Apr 5 '10 at 1:31
    
The remark looks too trivial to give much of anything. Note that the conclusion follows if e.g. you only assume that the range $\pi$ contains the constant functions. –  Bill Johnson Apr 5 '10 at 15:52
    
Maybe less trivial: it is true also when $\pi$ has a dense range. –  Ady Apr 6 '10 at 3:04

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